Errors in Ballentine (QM Textbook)?

[atyy]

He simple defines what is understood as a projective or von Neumann filter measurement. It's not a general rule or postulate but just a definition of a special type of experiment, which an be formulated in terms of the postulates of the minimal interpretation (as described in our Insights article). That such types of experiments are feasible in the real world is also evident from the many real-world experiments done with all kinds of systems in the labs where QT is investigated (e.g., quantum optics, AMO, HEP, condensed matter...).

So it's still a postulate. It's postulate 7 in https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/.

 

[vanhees71]

I'm not happy with calling it a postulate for the said reason. It's the definition of a special (usually idealized) kind of experiments. It's rather a question of how to apply the theory to a specific kind of preparation-observation procedures in each individual case of such a kind of experiment.

 

[atyy]

I'm not happy with calling it a postulate for the said reason. It's the definition of a special (usually idealized) kind of experiments. It's rather a question of how to apply the theory to a specific kind of preparation-observation procedures in each individual case of such a kind of experiment.

Well, that's an easily fixed reason. We can just use the more general state reduction postulate. For discrete variables, the more general state reduction postulate can be derived by using the projection postulate on a measurement model. This complaint is different from Ballentine's criticism of orthodox quantum mechanics by asserting that there is only unitary evolution of the quantum state.

 

[vanhees71]

There is only unitary evolution of the quantum state when considering a closed system.

A filter measurement necessarily involves more than the measured system, namely the filter.

 

[atyy]

There is only unitary evolution of the quantum state when considering a closed system.

A filter measurement necessarily involves more than the measured system, namely the filter.

Then after that, one needs the state reduction postulate.

 

[vanhees71]

No, you need to take a partial trace and describe the evolution by some master equation. That can be FAPP a kind of "state reduction", but it's nothing outside the dynamical laws of QT!

 

[atyy]

No, you need to take a partial trace and describe the evolution by some master equation. That can be FAPP a kind of "state reduction", but it's nothing outside the dynamical laws of QT!

Well, we shall have to disagree. There is a reason state reduction is stated in many good textbooks.

 

[vanhees71]

Well, I fail to see its necessity and why one should have, in the case of local relativistic QFT (the Standard Model!), a self-contradiction between the successful formalism and an unneeded statement.

 

[Demystifier]

There is only unitary evolution of the quantum state when considering a closed system.

But unitary evolution is deterministic. Does it mean that the quantum state of a closed system evolves deterministically and that there is no randomness in the quantum state of a closed system? But something does change randomly in a closed system, right? So does it mean that, in a closed system, there is something which is not the quantum state?

 

[vanhees71]

There is no randomness in the evolution of the quantum state at all. Why should there be? The Schrödinger equation is a perfectly deterministic equation for the wave function.

Quantum theory is probabilistic in its notion of the meaning of the state (Born's rule). Observables don't necessarily take determined values but this depends on the state the system is prepared in. That's why measuring an observable on an ensemble of equally prepared systems, which is not determined to have a certain value, results in a random-number distribution whose statistics is described by the probabilities given by the state the system is prepared in (via Born's rule).

I don't understand your last sentence. A system is described by an observable algebra (realized usually by a set of self-adjoint operators) on an appropriate Hilbert space. The properties of the system is described by the statistical operator, representing its state. What else should there be?

 

[Demystifier]

A system is described by an observable algebra (realized usually by a set of self-adjoint operators) on an appropriate Hilbert space. The properties of the system is described by the statistical operator, representing its state. What else should there be?

So there are two things, the state (which is deterministic) and the observables (which are random). I have two questions.

1) Do observables have random values when they are not measured?

2) If observables are random, how is it compatible with the Heisenberg picture where the observable operator has a deterministic evolution with time?

 

[vanhees71]

1) doesn't make sense, and it's not what quantum theory says. Quantum theory tells you the probability to find a certain value when measuring an observable, given the state the measured system is prepared in.

2) Quantum theory doesn't depend on the picture of time evolution used. The observable (probabilistic) predictions of quantum theory are always in the picture-independent matrix elements of the statistical operator,
$$\rho(t,o,o') \langle o,t|\hat{\rho}(t)|o,t' \rangle.$$
where $|o,t \rangle$ is a common eigenvector of a complete set of compatible observables $O$.

Both the equations of motion for the states (statistical operators) and the self-adjoint operators representing observables in an arbitrary picture of time evolution are of course deterministic.

 

[Demystifier]

1) doesn't make sense, and it's not what quantum theory says. Quantum theory tells you the probability to find a certain value when measuring an observable, given the state the measured system is prepared in.

Does quantum theory say anything about those values when they are not measured?

2) Quantum theory doesn't depend on the picture of time evolution used. The observable (probabilistic) predictions of quantum theory are always in the picture-independent matrix elements of the statistical operator,
$$\rho(t,o,o') \langle o,t|\hat{\rho}(t)|o,t' \rangle.$$
where $|o,t \rangle$ is a common eigenvector of a complete set of compatible observables $O$.

Both the equations of motion for the states (statistical operators) and the self-adjoint operators representing observables in an arbitrary picture of time evolution are of course deterministic.

So there are 3 things, not 2. The state (which is deterministic), the observable operator (which is also deterministic), and the value of the observable operator (which is random). Is that right?

 

[vanhees71]

Quantum theory predicts the values of observables when measured. Physics doesn't care about unobserved things.

An observable operator doesn't take values. It's a linear mapping $\mathcal{H} \rightarrow \mathcal{H}$.

I'm a bit puzzled why we are discussing these completely basic undisputed facts about QT all of a sudden.

 

[Demystifier]

Physics doesn't care about unobserved things.

So why do you care that conserved charge exists even when it is not measured?

 

[Demystifier]

I'm a bit puzzled why we are discussing these completely basic undisputed facts about QT all of a sudden.

That's my strategy of phishing, to catch you in an inconsistency.
Now I think I know what exactly is inconsistent in your interpretation, it's inconsistent double standards of relevancy.

 

[MathematicalPhysicist]

So why do you care that conserved charge exists even when it is not measured?

Some might say that an unobserved universe doesn't exist; for an observer to exists in the first place it or he needs a universe to exists in and for such a universe to exists it needs an observer in it that will observe/notice its existence.

But this is just philosophy...
Life goes in circles anyways.

 

[kith]

I'm not happy with calling it a postulate for the said reason. It's the definition of a special (usually idealized) kind of experiments. It's rather a question of how to apply the theory to a specific kind of preparation-observation procedures in each individual case of such a kind of experiment.

I have quoted the general version of the postulate in the other thread. Do you think the state-after-measurement rule (2.93) there shouldn't be included as part of a postulate because it can be derived from the other postulates? (I'm not sure if it makes sense to keep these two threads separate)

 

[PeterDonis]

There is only unitary evolution of the quantum state when considering a closed system.

The term "closed system", at least as it is used in the 7 Basic Rules Insights article, does not include any system on which a measurement is being made. So your statement here, while true, is irrelevant to what happens when a measurement is made, which is the case under discussion.

There is no randomness in the evolution of the quantum state of a closed system at all.

See my bolded addition above. With that addition, you are simply stating a property of unitary evolution of a closed system, as that term is defined above. But again, that property is not relevant to what happens when a measurement is made.

 

[PeterDonis]

Everyone, please bear in mind that this thread is about the specific discussion of errors in Ballentine, and that we are in the regular QM forum, where the accepted statement of the postulates of QM is that given in the 7 Basic Rules Insights article. We are investigating the possibility that the Ballentine reference in that article might need to be corrected, but that is not intended to open the doors to a general discussion of everyone's views of QM. Also please bear in mind that interpretation discussions belong in the interpretations subforum, not this one.

 

[PeterDonis]

I'm not sure if it makes sense to keep these two threads separate

This thread is specifically about Ballentine, as I noted in my previous post just now. And as I also noted in that post, the relevant version of any postulate for purposes of this thread is what is in the 7 Basic Rules Insights article, not any other source.

 

[PeterDonis]

Since there seems to be enough information indicating that we will need to make some corrections to the 7 Basic Rules Insights article, I have moved posts specifically on that topic to the discussion thread for the article, here:

https://www.physicsforums.com/threads/the-7-basic-rules-of-quantum-mechanics.971724/

Please move further discussion of what corrections need to be made to the Insights article to that thread.

 

[kith]

I think this is a separate question from the one I described above.

In principle, yes. In practice, part of the critique of the projection postulate is that it isn't general. This discussion has been going on at PF without resolution for a long time and currently, I think the best road to identify the core of the issue is the general case. I think this thread and it's spin-off have contributed quite a bit here, so in any case thanks for starting them.

Perhaps we need to either augment the article or do a follow-up article to cover how the rules need to be generalized to the POVM formalism. If there is interest in doing that, I'll start a separate thread on that topic (and post a link to it here).

I would appreciate this but I can only contribute limited time and limited expertise.

 

[atyy]

The term "closed system", at least as it is used in the 7 Basic Rules Insights article, does not include any system on which a measurement is being made. So your statement here, while true, is irrelevant to what happens when a measurement is made, which is the case under discussion.

I suspect that @vanhees71 refers to a closed system, because he believes that we can in principle include the observer and measurement apparatus in the quantum state, so that there is only unitary evolution. This is also my reading of what Ballentine means in his textbook, given his criticism of standard QM. I believe that postulating unitary evolution without state reduction is not correct unless one introduces additional postulates (eg. as attempted by many worlds, hidden variables, which also remain non-standard).

 

[PeterDonis]

In practice, part of the critique of the projection postulate is that it isn't general. This discussion has been going on at PF without resolution for a long time and currently, I think the best road to identify the core of the issue is the general case.

As I noted in post #82 a little bit ago, I have moved that discussion to the comment thread on the Insights article.

 

[PeterDonis]

I suspect that @vanhees71 refers to a closed system, because he believes that we can in principle include the observer and measurement apparatus in the quantum state, so that there is only unitary evolution.

But just including the observer and measurement apparatus is not enough. You also have to include the environment, which potentially can include the entire rest of the universe.

I think at this point things become highly interpretation-dependent.

This is also my reading of what Ballentine means in his textbook, given his criticism of standard QM.

I'm not sure Ballentine's viewpoint is that of "only unitary evolution", because he believes quantum measurements have single outcomes. You can't get single outcomes out of only unitary evolution. I'm not sure Ballentine is taking any of the alternative viewpoints you mention (many worlds, hidden variables, etc.), but it doesn't seem to me like he is taking an "only unitary evolution" viewpoint either.

 

[strangerep]

I'm not sure Ballentine's viewpoint is that of "only unitary evolution", because he believes quantum measurements have single outcomes.

Please give a specific reference that supports your account of what Ballentine supposedly believes.

[Sorry, but I don't like unsupported verbaling. In the spirit of the PF rules, such claims need to be supported by appropriate references.]

 

[PeterDonis]

Please give a specific reference that supports your account of what Ballentine supposedly believes.

Um, his entire textbook?

Seriously, I'm not sure where to start, since the assumption that individual measurements have single outcomes seems to me to be there in pretty much everything he says. Certainly it seems to be a necessary assumption of the ensemble interpretation that he explicitly adopts. If you really aren't seeing that when you read his textbook, then I can try to pick out particular passages that give me that impression.

 

[atyy]

I'm not sure Ballentine's viewpoint is that of "only unitary evolution", because he believes quantum measurements have single outcomes. You can't get single outcomes out of only unitary evolution. I'm not sure Ballentine is taking any of the alternative viewpoints you mention (many worlds, hidden variables, etc.), but it doesn't seem to me like he is taking an "only unitary evolution" viewpoint either.

When I mean his point of view is only unitary evolution, I mean only unitary evolution as part of the first 6 postulates in the 7 Basic Rules, ie. including the Born Rule but excluding state reduction, and probably without hidden variables. I think this is the most plausible reading of his text, because of what he says in the section "The measurement theorem for general states" which contains Eq 9.10 to 9.13 (interpretation apart, the mathematics is essentially the same as Zurek's Eq 1-5 and Eq 6, without Eq 7 in https://arxiv.org/abs/quant-ph/0306072). In this section Ballentine uses only unitary evolution, shows that state reduction is not the outcome of unitary evolution, and uses that an argument against accepting state reduction.

The Born rule gives single measurement outcomes, but it does not give quantum states corresponding to the single measurement outcomes, which is why I think Ballentine wrongly rejects the state reduction postulate without replacing it with anything else.

The other plausible reading, but I think less likely, is that Ballentine assumes hidden variables, since he refers to Einstein's Ensemble interpretation. But like you, I think this is not likely what he means (otherwise vanhees71 would not read Ballentine as a minimal statistical interpretation).

 

[PeterDonis]

That's precisely the question. Which additional postulates do you mean?

Since your post took this question far beyond just a question about Ballentine specifically, and well over the line into interpretation, I have moved it to the other thread in the interpretations forum where collapse is being discussed:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/post-6445336